The optimal quartic separation between randomized and quantum queries

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Shows that the universal bound \(R(f)=O((1+Q(f))^4)\) for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and quantum worst-case bit-query complexities with error at most 1/3; computation between queries is unrestricted.

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The paper states

Theorem 1.1

In A Nearly Quartic Separation Between Randomized and Quantum Query Complexity, 5 October 2026

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The optimal power is \alpha_{\mathrm{tot}}=4. More precisely, \R(f)=O((1+\Q(f))^4) for every total Boolean function. For each fixed integer k\ge2, there are positive constants c_k,C_k,b_k and total Boolean functions F_{k,m}, for all sufficiently large powers of four m, such that

\Q(F_{k,m})\le C_k\sqrt m\,(\log m)^{b_k}, \qquad \R(F_{k,m})\ge c_k\frac{m^{2-1/k}}{(\log m)^2}.

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A Nearly Quartic Separation Between Randomized and Quantum Query Complexity

5 October 2026, current edition.

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We construct total Boolean functions with a nearly quartic separation between bounded-error randomized and quantum query complexity. Writing these complexities as \(\mathrm R(f)\) and \(\mathrm Q(f)\), the examples rule out every universal bound \(\mathrm R(f)=O((1+\mathrm Q(f))^\alpha)\) with α < 4. Thus the known quartic upper bound has the optimal exponent, disproving the conjectured cubic bound. Both complexities count worst-case bit queries with error at most 1/3 on every input.

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seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
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27 references in A Nearly Quartic Separation Between Randomized and Quantum Query Complexity

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